API

balance!(A; scale=true, permute=true) => Abal, B::Balancer

Balance a square matrix so that various operations are more stable.

If permute, then row and column permutations are found such that Abal has the block form [T₁ X Y; 0 C Z; 0 0 T₂] where T₁ and T₂ are upper-triangular. If scale, then a diagonal similarity transform using powers of 2 is found such that the 1-norm of the C block (or the whole of Abal, if not permuting) is near unity. The transformations are encoded into B so that they can be inverted for eigenvectors, etc. Balancing typically improves the accuracy of eigen-analysis.

triangularize(S::Schur{T}) => Schur{complex{T})

convert a (standard-form) quasi-triangular real Schur factorization into a triangular complex Schur factorization.

eigvecs(S::Schur; left=false) -> Matrix
eigvecs(S::GeneralizedSchur; left=false) -> Matrix

Compute right or left eigenvectors from a Schur decomposition. Eigenvectors are returned as columns of a matrix, ordered to match S.values.

For standard Schur, the returned eigenvectors have unit Euclidean norm, and the largest elements are real.

For generalized Schur, each vector is scaled so the largest element vⱼ satisfies abs(real(vⱼ)) + abs(imag(vⱼ)) = 1 (a convention from LAPACK).

eigvalscond(S::Schur,nsub::Integer) => Real

Estimate the reciprocal of the condition number of the nsub leading eigenvalues of S. (Use ordschur to move a subspace of interest to the front of S.)

See the LAPACK User's Guide for details of interpretation.

eigvalscond(S::GeneralizedSchur, nsub) => pl, pr

compute approx. reciprocal norms of projectors on left/right subspaces associated w/ leading nsub×nsub block of S. Use ordschur to select eigenvalues of interest.

An approximate bound on avg. absolute error of associated eigenvalues is ϵ * norm(vcat(A,B)) / pl. See LAPACK documentation for further details.

subspacesep(S::Schur,nsub::Integer) => Real

Estimate the reciprocal condition of the separation angle for the invariant subspace corresponding to the leading block of size nsub of a Schur decomposition. (Use ordschur to move a subspace of interest to the front of S.)

See the LAPACK User's Guide for details of interpretation.

gschur!(A::StridedMatrix) -> F::Schur

Destructive version of gschur (q.v.).

gschur!(H::Hessenberg, Z) -> F::Schur

Compute the Schur decomposition of a Hessenberg matrix. Subdiagonals of H must be real. If Z is provided, it is updated with the unitary transformations of the decomposition.

gschur(A::StridedMatrix) -> F::Schur

Computes the Schur factorization of matrix A using a generic implementation. See LinearAlgebra.schur for usage.

ggschur!(A, B) -> S::GeneralizedSchur

Computes the generalized Schur factorization of matrix pair A,B using a generic implementation. Overwrites A and B.

geigen!(A, alg=QRIteration(); sortby=eigsortby) -> E::Eigen

Computes the eigen-decomposition of a Hermitian (or real symmetric) matrix A using a generic implementation. Currently alg may be QRIteration() or DivideAndConquer(). Otherwise, similar to LinearAlgebra.eigen!.

geigvecs(S; left::Bool = false)

Compute eigenvectors from a Schur (or generalized Schur) decomposition S. Left eigenvectors are produced if so specified, otherwise right ones.

gordschur!(F::Schur, select) -> Schur gordschur!(F::GeneralizedSchur, select) -> GeneralizedSchur

Reorder a Schur decomposition so that the eigenvalues and subspaces corresponding to indices set to true in the Boolean vector (or BitVector) select are first. Uses a generic implementation. C.f. LinearAlgebra.ordschur!.

IllConditionException

Exception thrown when argument matrix or matrices are too ill-conditioned for the requested operation. The index field may indicate the block where near-singularity was detected.

UnconvergedException

Exception thrown when an iterative algorithm does not converge within the allowed number of steps.

set_piracy!(::Bool)

Convenience wrapper for determining whether piratical LinearAlgebra methods will be implemented by GenericSchur for future Julia sessions. Uses Preferences.